1. Introduction

Results concerning the existence and properties of fixed points are known as fixed point theorems. The theory of fixed point became an important tool in non-linear functional analysis since 1930. It is used widely in applied mathematics. The existence and types of solution always help to give geometrical interpretation, to discuss the behavior and to check stability of the concern system. The famous Banach contraction principle says that “every contraction map from a complete metric space to itself has a unique fixed point”. Due to the wide importance and application of this principle, several authors generalized this principle using either different contractive conditions or space structure.

Further, the study of common fixed points of mappings satisfying certain contractive conditions has been reinvestigated extensively by many mathematicians. The fixed point theorems related to altering distances between points in complete metric space have been obtained initially by D. Delbosco in 1967, F. Skof in 1977, M.S. Khan, M. Swaleh and S. Sessa in 1984.

Recently, Huang Long-Guang et al. (2007) introduced the concept of cone metric spaces in which set of real numbers has been replaced by a real Banach space and a partial order has been defined with the help of a subset (called cone) of that real Banach space. As the set of real numbers is well ordered but the concerned Banach space is only partially ordered, so it is a task to extend the existing results in metric space to cone metric spaces if possible. In this paper we have established common fixed point results for cone metric spaces which generalize the existing results in metric spaces of Sastry et al.[9] and Pandhare et al.[4].

We now give some preliminaries about cone metric spaces given by Huang Long-Guang et al.[2].

Let be a real Banach space and be a subset of is called a cone if

(i) is closed, non-empty and

(ii) and non-negative real numbers

(iii) .

For a given cone we can define a partial ordering with respect to by if and only if will stand for and while will stand for where denotes the interior of .

The cone is called regular if every increasing and bounded above sequence in is convergent. Equivalently the cone is regular if and only if every decreasing and bounded below sequence is convergent.

Definition1.1[2] Let be a non-empty set. Suppose the mapping satisfies

(i) for all if and only if

(ii)

(iii)

Then is called a cone metric on and is called a cone metric space.

Definition 1.2[2] Let be a cone metric space, a sequence in and

For every with we say that is:

(i) a Cauchy sequence if there is a natural number such that for all

(ii) convergent to if there is a natural number such

that for all for some

is called a complete cone metric space if every Cauchy sequence in is convergent.

Definition 1.3 Let be the set of all continuous self maps of satisfying

(i) is monotone increasing

(ii)

Then it is called an altering distance function on the cone.

2. Main Results

In this section we obtain two fixed point results on a complete cone metric space generalizing Theorem 2 of Sastry and Babu[9] and Pandhare and Waghmode[4] in turn.

Theorem 2.1 Let be sequence of self maps on complete cone metric space Assume that

(i) There exist a in such that

for all and for all distinct where

(ii) There is a point such that any two consecutive members of the sequence defined by are distinct.

Then has a unique common fixed point in X. In fact is Cauchy and the limit of is the unique common fixed point of .

Proof : Let

Then

This implies that

i.e., i.e., where

By induction, we get

(1)

This implies that are decreasing and bounded below sequences in.

As P is regular, will converge and as n

Now

Therefore is a decreasing sequence in . As is regular.

Then So that hence

(2)

Now we show that is Cauchy in .

If it is not so then there is a and sequences such that and

Assume that for infinitely many.

Then for such we have

which is a contradiction because .

Hence for large

Consequently

Hence

i.e.,

i.e., implying that so

i.e.,

Therefore This is again a contradiction.

Hence is a Cauchy sequence in. As is complete, limit of exists. Let it be.

There is a sequence in N such that Otherwise for large, which is not the case, since consecutive terms are different. With this subsequence we have for any positive integer

Taking limit

Since 0 < b < 1, it follows that so that

This shows that y is a fixed point of for each m. Thus is a common fixed point for the sequence .

Now we show that the fixed point is unique. Let be another common fixed point of then

Remark: If we take metric as the usual metric and cone in our theorem then we get Theorem 2 of Sastry and Babu[9] as a corollary.

Now we give our next result where satisfies an additional property given by

Theorem 2.2 Let be a sequence of self maps on a complete cone metric space

Assume that

(i)There exist a in with (*) such that

for all in and for all distinct in, where

(ii) There is a point in such that any two consecutive members of the sequence defined by are distinct.

Then has a unique common fixed point in . In fact is Cauchy and the limit point of is the unique common fixed point of .

Proof : Write

From (i) and (ii) , we have

This implies that

i.e., i.e.,

By induction it follows that

(3)

So are decreasing and bounded below sequences in.

As is regular cone, will converge and

Now

Therefore is a decreasing sequence in. As is regular

Then hence

(4)

Now we show that is Cauchy in . If it is not so, then there is a and sequences in such that and

Assume that, for infinitely many

Then for such we have

as

which is a contradiction because .

Hence for large

Consequently

Hence

i.e., . This is again a contradiction.

Hence is Cauchy sequence in . As is complete, limit of exists. Let it be

There is a sequence such that Otherwise for large which is not the case, since the consecutive terms are different. With this subsequence we have for any positive integer

Taking limit as , we have

i.e.,

i.e., i.e.,

This shows that is a fixed point of . Thus is a common fixed point for the sequence. The uniqueness of the common fixed point can be shown easily.

3. Examples

Example 3.1: Let with usual metric. Define by

so that. Then satisfies the condition (i) with

Observe that, for any non-zero, the sequence defined by has all its elements distinct so (ii) also holds; thus hypothesis of Theorem2.1 is satisfied and is the unique common fixed point of .

Example 3.2: Let with usual metric. Define by

Define so that Then,

If we take

Then

Hence condition (i) of Theorem 2.2 is satisfied. Observe that for any non zero in, the sequence defined by has all its elements distinct so the condition (ii) of Theorem 2.2 also holds and is the unique common fixed point.

4. Conclusions

The results obtained in this work extends the common fixed point results in metric spaces of Sastry and Babu[9] and Pandhare and Waghmode[4] to cone metric space in a more general setting in context with the space structure equipped with a partial order.